Finite Group Databasebeta

← #29576137

Subgroups — #29576137

Order 15616.

450 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455880C211yesmaximal, characteristic
2#21454461C211yesmaximal, characteristic, complemented
3#21444410C211yesmaximal, characteristic, complemented
4#21444271C211yesmaximal, characteristic, complemented
5#21444259C211yesmaximal, characteristic, complemented
6#21443710C211yesmaximal, characteristic, complemented
7C61×#1278C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C4×#11758222C2211yescharacteristic
10#16840087—21no—
11#16840058C411yescharacteristic
12#16840053C411yescharacteristic
13#16839963C411yescharacteristic
14#16839958C411yescharacteristic
15#16839909—21no—
16C4×#11758190—21no—
17C4×D488—21no—
18#16839821—22no—
19#16839813—22no—
20#16839788—21no—
21#16839786C2211yescharacteristic
22#16839728—21no—
23C4×#11758194C2211yescharacteristic
24#16839501—22no—
25#16839494—21no—
26#16839491—21no—
27#16839491—22no—
28C61×#431C2211yescharacteristic
29C4×M16×C61C2211yescharacteristic
30C2×C4×C8×C61C2211yescharacteristic
31C2×#11758222—21no—
32C2×#11758222D411yescharacteristic
33C2×#11758190—21no—
34C2×D488—21no—
35C2×#11758194—21no—
36C2×#11758194—21no—
37#14519777C2×C411yescharacteristic
38#14519774C2×C411yescharacteristic
39#14519765—21no—
40#14519751—21no—
41#14519751—21no—
42C4×Dic122C2×C411yescharacteristic
43C4×Dic122D411yescharacteristic
44C4×D244—21no—
45C4×D244—21no—
46C4×D244C2×C411yescharacteristic
47C4×D244D411yescharacteristic
48C4×C22⋊D61—21no—
49C4×C22⋊D61—21no—
50#14519736D411yescharacteristic
51#14519734—21no—
52C42×D61—21no—
53#14519711—21no—
54#14519709—21no—
55C4⋊#92418—21no—
56C4⋊#92418—21no—
57#14519706—24no—
58#14519702—21no—
59#14519699—21no—
60#14519699—21no—
61#14519699—21no—
62#14519689—24no—
63#14519688D412yes—
64C4×#92418—21no—
65C2×M16×C61—21no—
66C22×C8×C61—21no—
67C2×C42×C61C2311yescharacteristic
68C4⋊C8×C61C2×C411yescharacteristic
69C4⋊C8×C61C2×C411yescharacteristic
70C22⋊C8×C61—21no—
71#97×C61—21no—
72C4×C8×C61—21no—
73C4×C8×C61—21no—
74C4×C8×C61—21no—
75C4×C8×C61D412yes—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61#4111yescharacteristic
78C22×C4×C61#4111yescharacteristic
79C2×C22⋊D61—21no—
80C2×C22⋊D61—41no—
81#11758222—41no—
82#11758222—41no—
83#11758222—81no—
84#11758222—81no—
85C2×D244—21no—
86C2×D244—21no—
87C2×D244—21no—
88C2×D244C22⋊C411yescharacteristic
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—41no—
93C2×Dic122—21no—
94C2×Dic122C22⋊C411yescharacteristic
95M16×C61—22no—
96M16×C61—41no—
97C2×C8×C61—21no—
98C2×C8×C61—21no—
99C2×C8×C61—22no—
100C2×C8×C61—41no—
101C2×C8×C61—41no—
102C2×C8×C61—41no—
103C42×C61C22×C411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61C2×D411yescharacteristic
106C42×C61C2×D411yescharacteristic
107C22⋊Dic61—21no—
108#11758198—21no—
109#11758198—21no—
110#11758198—41no—
111#11758198—41no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61C22⋊C411yescharacteristic
115C4⋊Dic61C22⋊C411yescharacteristic
116#11758196—21no—
117#11758196—21no—
118C4×Dic61—21no—
119C4×Dic61—21no—
120C4×Dic61—21no—
121#11758194—41no—
122#11758194—41no—
123#11758194—81no—
124C2×#92418—22no—
125C2×#92418—41no—
126D488—42no—
127#11758190—42no—
128C23×C61#12311yescharacteristic
129C22×D61—21no—
130C22×D61—41no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61C2×C22⋊C411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61#12011yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12111yescharacteristic
143C22⋊D61—42no—
144C22⋊D61—82no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—41no—
149D244—21no—
150D244—21no—
151D244—41no—
152D244—41no—
153D244—41no—
154D244—41no—
155D244—81no—
156C4×D61—42no—
157C4×D61—41no—
158C4×D61—41no—
159C4×D61—41no—
160C4×D61—41no—
161C4×D61—81no—
162C4×D61—81no—
163Dic122—21no—
164Dic122—21no—
165Dic122—81no—
166C8×C61—42no—
167C8×C61—42no—
168C8×C61—41no—
169C8×C61—41no—
170#92418—42no—
171#92418—81no—
172#9213—611nomaximal, Sylow
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61#38611yescharacteristic
177C2×D61—41no—
178C2×D61—41no—
179C2×D61—41no—
180C2×D61—81no—
181C2×D61—81no—
182C2×D61—81no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61#42111yescharacteristic
187C4×C61#42111yescharacteristic
188C4×C61#44311yescharacteristic
189C4×C61#44411yescharacteristic
190C4×C61#47111yescharacteristic
191C4×C61#47211yescharacteristic
192Dic61—41no—
193Dic61—41no—
194Dic61—41no—
195Dic61—81no—
196#3106—611no—
197#2922—611no—
198#1525—611no—
199#1492—611no—
200#1489—611no—
201#1299—611no—
202#1278D6111yescharacteristic, complemented
203C2×C61—41no—
204C2×C61#187611yescharacteristic
205C2×C61#189711yescharacteristic
206C2×C61#190711yescharacteristic
207D61—81no—
208D61—161no—
209#532—611no—
210C4×#41—611no—
211#482—1221no—
212#477—611no—
213#476—611no—
214#460—611no—
215#459—611no—
216#449—1221no—
217C4×SD16—1221no—
218C4×D8—1221no—
219#433—1222no—
220#431—1222no—
221#431C2×D6111yescharacteristic
222#424—1221no—
223#423—611no—
224#408—1221no—
225C4×M16—611no—
226C4×M16C2×D6111yescharacteristic
227C2×C4×C8C2×D6111yescharacteristic
228#329—1222no—
229Q8⋊C8—1221no—
230D4⋊C8—1221no—
231D4⋊C8—1222no—
232C61#921311yescharacteristic, complemented, Sylow
233C2×#41—611no—
234C2×#41—1221no—
235C2×SD16—1221no—
236C2×D8—1221no—
237C2×M16—21no—
238C2×M16—1221no—
239C2×M16—1221no—
240C22×C8—21no—
241C4⋊Q8—611no—
242#127—611no—
243#124—1221no—
244#121—1221no—
245#121—1221no—
246C4×Q8—611no—
247C4×Q8—611no—
248C4×D4—611no—
249C4×D4—611no—
250C4×D4—1221no—
251C4×D4—1221no—
252C4×D4—1221no—
253C4×D4—1221no—
254#117—611no—
255#117—1221no—
256C2×C42—1221no—
257C2×C42C22×D6111yescharacteristic
258#107—1221no—
259#106—1221no—
260C4⋊C8—1224no—
261C4⋊C8—1221no—
262C4⋊C8—1221no—
263C4⋊C8C4×D6111yescharacteristic
264C4⋊C8C4×D6111yescharacteristic
265#103—1221no—
266#102—1221no—
267#102—1221no—
268#102—1221no—
269C22⋊C8—21no—
270C22⋊C8—1224no—
271#97—21no—
272#97—612no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—1221no—
277C4×C8D24412yes—
278#41—4881no—
279#41—4881no—
280#41—2441no—
281#41—2441no—
282C2×Q8—611no—
283C2×Q8—1221no—
284C2×D4—611no—
285C2×D4—1221no—
286C2×D4—1221no—
287C2×D4—1221no—
288C2×D4—1221no—
289C2×D4—2441no—
290C22×C4—1221no—
291C22×C4—1221no—
292C22×C4—1221no—
293C22×C4—2441no—
294C22×C4#1175822211yescharacteristic
295C22×C4D4×D6111yescharacteristic
296C22×C4#1175822611yescharacteristic
297SD16—2442no—
298D8—2442no—
299M16—22no—
300M16—41no—
301M16—4881no—
302M16—2441no—
303M16—2441no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—1222no—
311C2×C8—2441no—
312C4⋊C4—611no—
313C4⋊C4—611no—
314C4⋊C4—1221no—
315C4⋊C4—1221no—
316C4⋊C4—1221no—
317C4⋊C4—1221no—
318C22⋊C4—1221no—
319C22⋊C4—1221no—
320C22⋊C4—1221no—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C42—1221no—
324C42—1221no—
325C42—1221no—
326C42C2×C4×D6111yescharacteristic
327C42C2×D24411yescharacteristic
328C42D4×D6111yescharacteristic
329C42D4×D6111yescharacteristic
330C23—1221no—
331C23—2441no—
332C23#1451976211yescharacteristic
333Q8—1221no—
334Q8—1221no—
335Q8—4881no—
336D4—1221no—
337D4—1221no—
338D4—4881no—
339D4—4882no—
340D4—2442no—
341D4—2441no—
342D4—2441no—
343D4—2441no—
344D4—2441no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—1221no—
354C2×C4—4881no—
355C2×C4—4881no—
356C2×C4—2442no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4C22⋊C4×D6111yescharacteristic
363C2×C4#1451973911yescharacteristic
364C2×C4C4×D24411yescharacteristic
365C2×C4#1451974711yescharacteristic
366C2×C4#1451975011yescharacteristic
367C2×C4#1451975211yescharacteristic
368C8—42no—
369C8—42no—
370C8—41no—
371C8—41no—
372C8—4881no—
373C8—2442no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—4881no—
378C22—4881no—
379C22—4881no—
380C22—2441no—
381C22—2441no—
382C22—2441no—
383C22#1683964011yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—4881no—
388C4—2441no—
389C4—2441no—
390C4—2441no—
391C4#1683977711yescharacteristic
392C4#1683977711yescharacteristic
393C4#1683988011yescharacteristic
394C4#1683988511yescharacteristic
395C4#1684002611yescharacteristic
396C4#1684003311yescharacteristic
397C2—41no—
398C2—9761no—
399C2—4881no—
400C2#2144614511yescharacteristic
401C2#2144626611yescharacteristic
402C2#2144632611yescharacteristic
subgroupG/H
centerC2×C4#14519752
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1278C2
socleC22×C61#386
prime radical#21455880C2
cosocleC2×C42×C61C23
hypercenter#1278D61
hypocenterC61#9213
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21454461C2semidirect
3. #21444410C2semidirect
4. #21444271C2semidirect
5. #21444259C2semidirect
6. #21443710C2semidirect
7. C61×#1278C2semidirect
202. #1278D61semidirect
232. C61#9213semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214558802
#214544612
#214444102
#214442712
#214442592
#214437102
C61×#12782
#921361